Questions tagged [prime-numbers]

Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.

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104 views

Variants of Nicholson's inequalities for prime numbers, involving the Lambert $W$ function

The purpose of this post is ask about two closely related/inspired conjectures from inequalities due to Nicholson (see [1]) and Visser [2]. If my reasonings are right should be stronger versions of ...
0 votes
0 answers
128 views

A quadratic trinomial that generates only prime numbers of the form $4m+1$

It is known that Euler's polynomials $\,n^2+n+p\,$ ($p\,$ prime) represent a prime for $\,n=0,\,...,\,p-2\,$ if and only if the field $\,Q (\sqrt{1-4p})\,$ has class number $\,h=1$. The best ...
6 votes
1 answer
227 views

Is there a connection between the average 'compositeness' of a rational number and $\phi$ (golden ratio)?

Let $n\in N$, where $n = p_{1}^{k_{1}}p_{2}^{k_{2}}...p_{m}^{k_{m}}$ for $p_{i}$ prime. Define the 'density' of $n$ as: $d(n) = \frac{(p_{1}+1)^{k_{1}}(p_{2}+1)^{k_{2}}...(p_{m}+1)^{k_{m}}}{n}$ ...
0 votes
1 answer
159 views

does the ratio of the count of rational numbers on an $n\times n$ grid to $n^2$, converge as $n$ tends to infinity [closed]

Suppose we order the rational numbers using the diagonal method (used to prove they are countable) using an $n\times n$ grid. Now suppose we count the distinct rational numbers (those points on the ...
0 votes
1 answer
237 views

What can this $\int_{0}^{t} (\pi(x)-Li(x)) dx$ tell us about primes distribution?

Many papers I have read which are related to primes distribution only it discussed sign and refinement Bounds of $\pi(x)-Li (x)$ with $\pi(x)$ is a prime counting function and $Li (x)$ is the ...
1 vote
1 answer
188 views

Resolution of an inequality on integers

I’m trying to resolve respect to $k$ the following inequality, $$ k\left(\log k +\log \log k-\alpha+O\left(\frac{\log \log k}{\log k}\right)\right)\geq x, $$ in order to obtain, under the condition $...
1 vote
0 answers
142 views

About the distribution of Fibonacci numbers that are primes

Let's consider the Fibonacci sequence, that is the sequence of naturals defined by: $F_1=F_2=1$ $F_{n+1}=F_{n}+F_{n-1}$ It is an open problem whether the sequence contains an infinite number of ...
4 votes
0 answers
143 views

Moments of the prime counting function given the moments of the second Chebyshev function

I have read this article (Montgomery and Soundararajan: Primes in short intervals. http://arxiv.org/abs/math/0409258 ). In the second page of the article, it is stated that the mean and variance of $\...
1 vote
0 answers
175 views

Given a prime $\,p\ne3$, is it always possible to find another prime q such that $\,\phi(q)=\phi(p\cdot2^{2n+1})\,$ for some $n\in\mathbb{N}$?

Given a prime $\,p\ne3$, is it always possible to find another prime q such that $\,\phi(q)=\phi(p\cdot2^{2n+1})\,$ for some $n\in\mathbb{N}$ ($\,\phi\,$ is the Euler's totient function)? Some ...
3 votes
1 answer
310 views

Why is this sequence a good prime-generator?

For $n \in \mathbb N$ we can observe the $n$ remainders $b_1,...,b_n$ by writing $n$ as $n=a_k \cdot k+b_k$ for $1 \leq k \leq n$. Because of the familiar division-with-remainder theorem we have $0 \...
11 votes
1 answer
421 views

How many numbers $\le x$ can be factorized into three numbers which form the sides of a triangle?

Note: Posting in MO since it was unanswered in MSE Definition: We say that a natural number $n$ has triangular divisors if it has at least one triplet of divisors $n = d_1d_2d_3, 1 \le d_1 \le d_2 \...
-1 votes
1 answer
281 views

Is it possible to determine whether the sequence $\,a_0=p,\;a_{n+1}=(a_n-2)\cdot a_n+2\,$ will reach another prime number?

Given a prime $\,p\,$ let's consider the following sequence: $a_0=p$ $a_{n+1}=(a_n-2)\cdot a_n+2$ Is it possible to determine whether the sequence $\,a_n\,$ will reach, sooner or later, another ...
1 vote
0 answers
277 views

Prime numbers in this region

Let $q \geq 5$ be a prime number, and consider : $N_q = \displaystyle{\small \prod_{\substack{p \leq q \\ \text{p prime}}} {\normalsize p}}$ Using Chinese remainder theorem we can show that : $$\#\{(...
0 votes
0 answers
141 views

Given $\,m=\prod_k {p_k}^{\alpha_k}\,$ and the function $\,g(m)=\sum_k \alpha_k(p_k-1)^2$, find all solutions of the equation $\,g(2n)=n$

Let's consider the unique decomposition of a natural number $\,m\,$ into its prime factors: $$\prod_k {p_k}^{\alpha_k}$$ Then, let's define the following arithmetic function (completely additive) $\,g:...
3 votes
1 answer
266 views

Solutions in primes of the equation $\,3p^2+q^2=r^2+3$

Let's consider the Diophantine equation $\,3p^2+q^2=r^2+3$. Actually, I am interested only in the solutions represented by sets $\,(p,q,r)\,$ of prime numbers. It's easy to prove that if $\,(p,q)\,$ ...
13 votes
1 answer
1k views

About the number of primes which are the sum of 3 consecutive primes (OEIS A034962)

I made some numerical simulations about the number of primes which are the sum of 3 consecutive primes (OEIS A034962), that is for instance: $$5+7+11=23$$ $$7+11+13=31$$ $$11+13+17=41$$ $$17+19+23=59$$...
15 votes
1 answer
1k views

Greatest prime factor of n and n+1

For a positive integer $n$ we denote its largest prime factor by $\operatorname{gpf}(n)$. Let's call a pair of distinct primes $(p,q)$ $\textbf{nice}$ if there are no natural numbers $n$ such that $\...
1 vote
1 answer
116 views

Sequences of positive integers $(a_{k})_{k \in \omega}$ that only give finitely many zeros modulo $p_{k}$ in total for all polynomials

Let $(a_{k})_{k \in \omega}$ be a sequence of positive integers such that $a_{k} < p_{k}$, $a_{k} \leq a_{k+1}$ and $\lim_{k \rightarrow \infty} a_{k}=\infty$ where $p_{k}$ is the k-th prime ...
2 votes
0 answers
170 views

Funny questions about Moebius Function

I need to firstly claim that my research is not about number theory, however, I am pretty interested in it, especially funny questions in number theory, e.g. Kollatz Conjecture. Three years ago, I ...
0 votes
0 answers
77 views

Construction of (general class of) function(s), which sieves out primes, w.r.t. given conditions:

Consider the function $F(x)$ defined in following manner: $F(n)$ is finite (likely $F(x)\in[0,1]$) if $n$ is prime and zero otherwise: It has to satisfy following conditions: (1) $F(x)$ is ...
2 votes
1 answer
321 views

Can someone explain some of the steps in this paper clearly?

I'm having trouble understanding the steps this paper makes to come to the conclusion $p_{f}(d) \sim e^d\sqrt{d}$ Marek Wolf, First occurrence of a given gap between consecutive primes, preprint, ...
0 votes
1 answer
352 views

A sufficient condition for a set of primes to be the set of reducibility of an integer polynomial

Let $P$ be the set of all positive primes. Let $S$ an arbitrary infinite subset of $P$ satisfying the following assumption: there exists a finite Galois extension $K$ of $\mathbb{Q}$ and a conjugacy ...
2 votes
1 answer
525 views

Sets of primes with a given Frobenius conjugacy class

Let $a$ and $b$ be two relatively prime positive integers. Denote by $S$ the set of all positive primes of the form $a+bn$ (where $n$ is an integer, possibly zero or negative). Let $S'$ be any set of ...
4 votes
1 answer
313 views

Estimating certain short Kloosterman sums

Recall that for the classical Kloosterman sum $$ K(a,b,p^t):= \sum_{x \in (\mathbb{Z}/ p^t \mathbb{Z})^* } \psi \left(\frac{ax+bx^{-1}}{p^t} \right),$$ where $\psi(x)=e^{2\pi ix}$, $a,b,t$ are natural ...
2 votes
1 answer
161 views

Numerical estimates for a function relating to twin primes :

Consider the following function : $$F(s)= \sum_{\text{$p,\ p+2$ are primes}} \left({\frac{1}{p^s}}+{\frac{1}{(p+2)^s}}\right).$$ Brun's theorem tells us that $F(1)$ is finite. We are looking for ...
1 vote
2 answers
141 views

Is the abscissa of convergence of $s\mapsto\sum_{n>0}(ng_{n}/2)^{-s}$ known?

The famous Polignac conjecture posits that the $n$-th prime gap $g_{n}:=p_{n+1}-p_{n}$ attains all even positive integral values infinitely many times, which implies $\displaystyle{\zeta_{Pol}:=s\...
4 votes
1 answer
819 views

Primality test similar to the AKS test

Let us define polynomials $P_n^{(a)}(x)$ as follows : $P_n^{(a)}(x)=\left(\frac{1}{2}\right)\cdot\left(\left(x-\sqrt{x^2+a}\right)^n+\left(x+\sqrt{x^2+a}\right)^n\right)$ We can define these ...
2 votes
0 answers
156 views

Questions about a certain sequence of naturals generated by primorials

I'm working on the following sequence of naturals (which is NOT listed in OEIS) $$3,5,11,17,23,29,59,89,119,149,179,209,419,629,839,1049,1259,1469,1679,...$$ whose elements are generated this way $$3=(...
-1 votes
1 answer
242 views

A conjecture about an inequality that involve Ramanujan primes

In this post we denote for integers $n\geq 1$ the $n$-th Ramanujan prime as $R_n$ (thus the sequence A104272 from the On-Line Encyclopedia of Integer Sequences), I add a conjecture that I think can be ...
4 votes
2 answers
1k views

Calculating the infinite product from the Hardy-Littlewood Conjecture F

The Hardy-Littlewood Conjecture F [1] involves the infinite product $$\prod\left(1-\frac{1}{\varpi-1}\left(\frac D\varpi\right)\right)$$ where $\varpi$ ranges over the odd primes and $\left(\frac D\...
2 votes
0 answers
119 views

On the set $\{n>0:\ n\ \text{is a quadratic nonresidue modulo the}\ n\text{th prime}\}$

Let $S$ denote the set of positive integers $n$ with $n$ a quadratic nonresidue modulo the $n$th prime $p_n$. The first 20 elements of $S$ are $$2,\, 3,\, 6,\, 7,\, 8,\, 10,\, 11,\, 13,\, 15,\, 18,\, ...
0 votes
1 answer
132 views

A density zero set of primes dividing the values of a non-constant integer polynomial

For a given $P\in \mathbb{Z}[x]$ call a positive prime $p$ good if there exists $n\in \mathbb{Z}$ such that $p$ divides $P(n)$. Does there exist a non-constant $P$ such that the set of good primes has ...
5 votes
1 answer
164 views

Representation of primes of the form $4m+3$ with double radicals

Let $\,q\,$ be a prime of the form $\,4\, m_q+3$. I ask if it is always possible to find two primes $\,p_1$ and $\,p_2$ of the form $\,4\, m_p+1$ such that $$q=\sqrt{p_1+\sqrt{p_2+q}}$$ E.g. $$3=\...
2 votes
1 answer
289 views

Convergence of series $\sum_{k=1}^{\infty}\frac{p_{k+1}-p_k}{(p_{k+1}+p_k)^\alpha}$

I ask if the series $$\sum_{k=1}^{\infty}\frac{p_{k+1}-p_k}{(p_{k+1}+p_k)^\alpha}$$ where $p_k$ stands for the prime of index $k$, has the same properties of convergence of the series $$\sum_{k=1}^{\...
6 votes
0 answers
197 views

Smooth integers with lower bound on $\omega(n)$

Define $(b,c)$-smooth integers to be integers having all prime factors bigger than $c$ and smaller than $b$. Probability a number is $(b,1)$-smooth is governed by the Dickman function while ...
-3 votes
1 answer
178 views

Asymptotic behavior of $\sum_{k=1}^{n}\frac{p_{k+1}}{p_{k+1}-p_k}$

I refer to my previous question Asymptotic behavior of a certain sum of ratios of consecutives primes. We can split the sum $$\sum_{k=1}^{n}\frac{p_{k+1}+p_k}{p_{k+1}-p_k}$$ where $p_k$ stands for the ...
10 votes
1 answer
465 views

Asymptotic behavior of a certain sum of ratios of consecutives primes

I am looking for the asymptotic growth of the following sum $$\sum_{k=1}^{n}\frac{p_{k+1}+p_k}{p_{k+1}-p_k}$$ where $p_k$ stands for the prime of index $k$. Manual computations show, for small values ...
6 votes
3 answers
821 views

Is it possible to multiply two series to get as a result all composite numbers?

I was toying with the following problem: Is it possible to find two infinite integer sequences $(a_n), (b_n)>0$ such that $\sum_{n=1}^{\infty}\frac{1}{(a_n)^s}\cdot \sum_{n=1}^{\infty}\frac{1}{(b_n)...
7 votes
1 answer
220 views

The asymptotic of $|\{1\leq n\leq x|\gcd(n,S(n))=1\}|$, with $S(n)$ the sum of remainders, and get idea for other miscellany problem

Let $n\geq 1$ be an integer. In this post we denote the sum of remainders function as $$S(n)=\sum_{k=1}^n n \bmod k,$$ for example $S(1)=S(2)=0+0$ and $S(5)=0+1+2+1+0=4$. In the literature there are ...
4 votes
0 answers
141 views

Can this number be interpreted as a fractal dimension?

Under Goldbach's conjecture, let's denote for a large enough integer $n$ by $r_{0}(n)$ the quantity $\inf\{r>0,(n-r,n+r)\in\mathbb{P}^2\}$ and by $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n+r_{0}(n))$. Let's ...
0 votes
0 answers
64 views

Can power of a prime number be approximated by product of powers of two adjacent prime numbers?

With prime numbers $a < b < c$ and no primes exist in ranges $(a, b)$ and $(b, c)$, is it possible that there exists positive integers $x$, $y$, $z$ such that $|a^x c^z-b^y|=2$?
12 votes
0 answers
2k views

Would the following conjectures imply $\lim\inf_{n\to\infty}p_{n+k}-p_{n}=O(k\log k)$?

Assume Goldbach's conjecture. Then for every $n\ge 2$ there exists at least one non-negative integer $r\le n-2$ such that both $n+r$ and $n-r$ are primes. Let's write $r_{0}(n):=\inf\{r\le n-2, (n-r,n+...
5 votes
0 answers
136 views

Is finding positive integer solutions of $\zeta(a/b) = c$ equivalent to deciding the rationality of $\gamma$?

This question requires little bit of explanation of the background hence it is a bit lengthy. Note: The question was initially posted in MSE but did not get answers hence posting in MO. For every ...
4 votes
0 answers
139 views

Is it true that $|\{k^{k+1}+(k+1)^k\pmod p:\ k=0,\ldots,p-1\}|=(1-e^{-1})p+O(\sqrt{p})\ ?$

For each prime $p$, let us define $$w_p:=|\{k^{k+1}+(k+1)^k\pmod p:\ k=0,\ldots,p-1\}|,$$ where $a\pmod p$ denotes the residue class $a+p\mathbb Z$. Based on my computation, I conjecture that $$w_p=...
4 votes
0 answers
163 views

Primitive roots modulo primes related to Fibonacci numbers or Lucas numbers

The Fibonacci numbers $F_0,F_1,F_2,\ldots$ and the Lucas numbers $L_0,L_1,L_2,\ldots$ are given by $$F_0=0,\ F_1=1,\ \text{and}\ F_{n+1}=F_n+F_{n-1}\ (n=1,2,3,\ldots)$$ and $$L_0=2,\ L_1=1,\ \text{...
12 votes
2 answers
1k views

$Ax^2 + By^3$ representing infinitely many primes

Are there any known results of the form there are infinitely many primes of the form $Ax^2 + By^3$ for integers $A$, $B$? Assuming there are currently no known results of this form, what is the ...
6 votes
0 answers
421 views

Average value of $\prod_{p|d}{p-1\over p-2}$ for $d=nq$, $n\in{\mathbb N}$, with $p$ odd prime

$\newcommand{\mean}{\mathop{\mathrm{mean}}}$ Define $$ S(d) = \prod_{p|d\atop p>2}{p-1\over p-2}. $$ Bombieri and Davenport (1966) proved that $$ \mean\limits_{d\in{\mathbb N}} S(d) = \mean\...
5 votes
0 answers
612 views

is there a link with the probabilistic model for prime numbers?

Let $x \in \mathbb{R}_+$ and $k \in \mathbb{N}^{*}$. Let : $$\mathcal{A}(x)=\#\{(a_1, a_2, \ldots, a_k) \in \mathbb{P}^k \mid (a_1, a_2, \ldots, a_k \text{ verifying some properties}) \, , a_k \...
4 votes
1 answer
875 views

Least Prime Factors: found a counting formula for a given range -- what is the standard approach?

Hi Everyone, I am a math amateur who for the past year has been working on better understanding Bertrand's Postulate, the Ramanujan Primes, and the recent expansion of Bertrand's Postulate (always a ...
1 vote
1 answer
213 views

Prime analogue of Champernowne's constant

Are any non-trivial properties known about the constant 0.2357111317192329... that is obtained by catenating the digits of sequence of prime numbers in base 10 or in other bases, especially whether it ...

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